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arXiv · 2203.04801

Extensions of valuations to rational function fields over completions

Arpan Dutta
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Abstract

Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of $v$ to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$. The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.

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Arpan Dutta. 2022-03-09. Extensions of valuations to rational function fields over completions. https://arxiv.org/abs/2203.04801

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